Scaling limits of looperased random walks and uniform spanning trees
 Oded Schramm
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The uniform spanning tree (UST) and the looperased random walk (LERW) are strongly related probabilistic processes. We consider the limits of these models on a fine grid in the plane, as the mesh goes to zero. Although the existence of scaling limits is still unproven, subsequential scaling limits can be defined in various ways, and do exist. We establish some basic a.s. properties of these subsequential scaling limits in the plane. It is proved that any LERW subsequential scaling limit is a simple path, and that the trunk of any UST subsequential scaling limit is a topological tree, which is dense in the plane.
The scaling limits of these processes are conjectured to be conformally invariant in dimension 2. We make a precise statement of the conformal invariance conjecture for the LERW, and show that this conjecture implies an explicit construction of the scaling limit, as follows. Consider the Löwner differential equation
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 Title
 Scaling limits of looperased random walks and uniform spanning trees
 Journal

Israel Journal of Mathematics
Volume 118, Issue 1 , pp 221288
 Cover Date
 20001201
 DOI
 10.1007/BF02803524
 Print ISSN
 00212172
 Online ISSN
 15658511
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Industry Sectors
 Authors

 Oded Schramm ^{(1)}
 Author Affiliations

 1. Department of Mathematics, The Weizmann Institute of Science, 76100, Rehovot, Israel